The potential communication capacity of optical fibers operating in the low loss wavelength windows of 1.3 .mu.m and 1.5 .mu.m is in the order of tens of Terahertz. The practical utilization of this bandwidth may be realized through the use of wavelength division multiplexing (WDM). In this scheme the spectral range is subdivided and allocated to different carrier wavelengths (channels) which are multiplexed onto the same fiber. The frequency bandwidth that an individual channel occupies depends on a number of factors, including the impressed modulation bandwidth, margins to accommodate for carrier frequency drift and carrier frequency uncertainty, and also to reduce possible cross-talk between channels.
Although the isolated optical fiber may inherently have tremendous information carrying capacity, the overall optical communication link may be significantly restricted in bandwidth. These restrictions may result from the limited optical amplifier spectral windows, the availability of lasing sources and their tuning ranges, and also filter tuning ranges. Hence, to achieve efficient use of bandwidth requires that the available communications windows be densely filled with multiplexed channels. At the output of such a system, filters are needed to separate the wavelength channels. The performance of these wavelength filters in their ability to reject out of band signals, critically determines channel spacing and hence channel density.
The type of filters of interest here operate through a wavelength dependent exchange of power between two waveguide modes. It is well known that two waveguides placed in close proximity may exchange power through their evanescent fields, which penetrate the guiding layer of the other waveguide. This power exchange occurs continuously along the propagation direction, with a rate determined by the inter-waveguide spacing and the degree of velocity-matching of the two modes. If the velocities of the two waveguide modes are identical, the situation is termed `synchronous` or `phase matched`, and the power coupled into one of the waveguides accumulates constructively. Complete power exchange is then possible and occurs at a characteristic coupling length L.sub.c which is determined by the structure of the device. If the modes propagate at different velocities, then this condition is termed `non-phase matched` or `asynchronous`. The power in the coupled waveguide accumulates with a phase error, leading to incomplete power transfer in this case. The larger the phase-mismatch, the faster the phase-error accumulates, which results in less power being transferred to the coupled waveguide.
FIG. 1a shows two typical waveguides placed in parallel, a configuration known as the directional coupler. The input power is initially launched into waveguide 1 (the `input` guide), and the output is extracted from guide 2 (the `coupled` guide). FIG. 1b shows the power in waveguide 2 (the `coupled` guide) as a function of device length for two cases of phase-matching. The solid curve represents a synchronous case, with complete power exchange occurring at L.sub.c =5 mm. The dashed curve represents coupling between asynchronous modes.
Wavelength selectivity in the directional coupler occurs through differential velocity dispersion. At the design wavelength, the velocities of the two modes are equal. As the wavelength is changed or `detuned`, the mode velocities necessarily change. For filtering action however, it is critical that the difference between these mode velocities changes, i.e., a differential velocity dispersion is required. The rate of change of differential velocity with respect to wavelength is the primary factor in determining filter bandwidth. This rate is a function of material type and waveguide structure. FIG. 2 shows the filter response of a parallel directional coupler. The abscissa is in terms of a normalized detuning factor, .DELTA..beta., which is a measure of the velocity difference between the coupled modes. This axis can be converted into an actual wavelength scale when the relationship between .DELTA..beta. and wavelength .lambda., is established for a particular device. The ordinate is the power in the coupled waveguide, in logarithmic scale, for a device of fixed length. The normalized half-power bandwidth is 8.4 radians (rad), and the maximum sidelobe level is at -9.3 db.
For optical communications purposes, a sidelobe level of -9.3 db is too large, since it would represent a significant cross-talk to an adjacent wavelength channel, if these channels were spaced by the width of the passband. If it is required that the cross talk in an adjacent channel be less than -9.3 db, the spacing between adjacent channels in the wavelength domain must be made much larger than the main passband width. Since the sidelobe levels decrease at a slow rate with detuning (see FIG. 2), the channels must be widely separated. Hence, a severe penalty is paid in terms of channel density, and hence information carrying capacity, for the price of low cross-talk. It is very desirable then to identify some degree of freedom which may be used to improve filter response.
The degree of freedom most commonly used in directional couplers is a modulation of the interaction strength of the two coupled modes. This may be achieved for example, by modulating the inter-waveguide separation in the directional coupler. In many other branches of optics and physics this process is known as `apodization`. In waveguide theory it is referred to as `tapering`
Directional coupler devices are commonly modeled through a set of coupled differential equations such as: ##EQU1## In (1a) and (1b) A.sub.1 and A.sub.2 represent the amplitudes in waveguides 1 and 2. .DELTA..beta. is the detuning constant and k is the interaction strength. k depends on the waveguide structure and is strongly influenced by the separation between adjacent waveguides. The physical origin of this coupling may be due to the interaction of evanescent fields in a uniform coupler, (as in FIG. 1a), or coherent scattering in a grating-assisted coupler.
By varying the interaction strength k along the directional coupler, the spectral response of the device may be improved. The physical origin of this improvement is in the interferometric nature of the coupling process: at every position along the coupler, power is being transferred from the input waveguide to the coupled waveguide. The total power in the coupled waveguide at some point then, is an interferometric sum of all the power coupled into the waveguide prior to that point. That is, a sum including relative phase delays. By adjusting the interaction strength k along the waveguide, one dictates the rate of power transfer at each position, along with its phase relationship to the total coupled power. By judicious choice of the coupling taper shape k(z), it is theoretically possible to generate any (passive) response.
How to calculate the taper shape k(z) for a desired response has been a long standing unanswered design question. The original proposal, suggested in 1978 by Alferness et el, IEEE J. Quantum Physics, QE-14, No. 11, 1978, pp. 843-847, was based on an approximate Fourier transform relation which gave a few promising shapes. This proved to be a useful guideline in improving actual device response (Alferness, Applied Physics Letters, Vol. 35, No. 3, 1979, pp. 260-262), but thus far has been unsuccessful in yielding the sidelobe suppression required in communication systems. To date no improvements have been advanced, and the same shapes suggested in 1978 continue to be the only ones analyzed (see, for example, H. Sakata, Optical Letters, Vol. 17, No. 7, 1992, pp. 463-465).
The goal of filter design is to solve for the interaction function k(z) of (1), given some desired output response as a function of wavelength A.sub.2 (.lambda.). However, when k(z) is non-constant, the set of coupled equations in (1) has no analytic solution in general. Hence, filter design is currently guided by a set of approximate solutions. The most important of these approximate solutions is the Fourier transform relation, given by: EQU A.sub.2 (.DELTA..beta.).apprxeq..intg.k(z)e.sup.-j.DELTA..beta..multidot.z dz Equation (2)
In Equation (2), A.sub.2 (.DELTA..beta.) is the amplitude in the output or coupled waveguide as a function of detuning .DELTA..beta., (which may be related to the actual wavelength .lambda.). Because Equation (2) represents a Fourier transform relation between k(z) in the spatial domain and A.sub.2 (.DELTA..beta.) in the wavelength domain, the principle of duality may be used. That is, given a desired A.sub.2 (.DELTA..beta.), k(z) is found by the inverse Fourier transform. This approximation is valid for small coupling values, and does not extend to describe the critical region of the main passband and first few sidelobes. No analytic solution currently treats the important region around the central wavelength.